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SDP hierarchies and quantum states
Aram Harrow (MIT)
Simons Institute 2014.1.17
a theorem
Let M2R+m£n.
Say that a set S⊆[n]k is δ-good if ∃φ:[m]k  S
such that ∀(j1, …, jk)∈S,
f(k,δ):= max{ |S| : ∃S⊆[n]k, S is δ-good}
Then
a theorem
The capacity of a noisy channel equals the
maximum over input distributions of the mutual
information between input and output.
[Shannon ’49]
2->4 norm
Define ||x||p := (𝔼i |xi|p)1/p
Let A2Rm£n.
||A||2!4 := max { ||Ax||4 : ||x||2 = 1}
How hard is it to estimate this?
optimization problem over a degree-4 polynomial
SDP relaxation for 2->4 norm
where
L is a linear map from deg ≤k polys to R
L[1] = 1
L[p(x) (𝔼i xi2 – 1)] = 0 if p(x) has degree ≤ k-2
L[p(x)2] ≥ 0
if p(x) has degree ≤ k/2
Converges to correct answer as k∞. [Parrilo ’00, Lasserre ‘01]
Runs in time nO(k)
Why is this an SDP?
Constraint: L[p(x)2] ≥ 0 whenever deg(p) ≤ k/2
p(x) = ∑α pα xα
α= (α1 , …, αn)
αi ≥ 0
∑iαi ≤ k/2
L[p(x)2] = ∑α,β L[xα+β] pα pβ
≥0 for all p(x) iff
M is positive semi-definite (PSD),
where Mα,β = L[xα+β]
Why care about 2->4 norm?
Unique Games (UG):
Given a system of linear equations: xi – xj = aij mod k.
Determine whether ≥1-² or ≤² fraction are satisfiable.
Small-Set Expansion (SSE):
Is the minimum expansion of a set with ≤±n vertices ≥1-² or ≤²?
UG ≈ SSE ≤ 2->4
G = normalized adjacency matrix
Pλ = largest projector s.t. G ≥ ¸P
Theorem:
All sets of volume ≤ ± have expansion ≥ 1 - ¸O(1)
iff
||Pλ||2->4 ≤ 1/±O(1)
quantum states
Pure states
• A quantum (pure) state is a unit vector v∈Cn
• Given states v∈Cm and w∈Cn, their joint state is
v⊗w∈Cmn, defined as (v⊗w)i,j = vi wj.
• u is entangled iff it cannot be written as u = v⊗w.
Density matrices
• ρ satisfying ρ≥0, tr[ρ]=1
• extreme points are pure states, i.e. vv*.
• can have classical correlation and/or quantum entanglement
correlated
entangled
when is a mixed state
entangled?
Definition: ρ is separable (i.e. not entangled)
if it can be written as
ρ = ∑i pi vi vi* ⊗ wi wi*
Sep = conv{vv* ⊗
ww*}
probability
distribution
unit vectors
Weak membership problem: Given ρ and the promise that
ρ∈Sep or ρ is far from Sep, determine which is the case.
Optimization: hSep(M) := max { tr[Mρ] : ρ∈Sep }
monogamy of entanglement
Physics version: ρABC a state on systems ABC
AB entanglement and AC entanglement trade off.
“proof”: If ρAB is very entangled, then measuring B can
reduce the entropy of A, so ρAC cannot be very entangled.
Partial trace: ρAB = trC ρABC
Works for any basis of C. Interpret as different choices
of measurement on C.
A hierachy of tests for
entanglement
Definition: ρAB is k-extendable if there exists an extension
with
for each i.
all quantum states (= 1-extendable)
2-extendable
100-extendable
separable =
∞-extendable
Algorithms: Can search/optimize over k-extendable states in time nO(k).
Question: How close are k-extendable states to separable states?
2->4 norm ≈ hSep
A = ∑i ei a iT
Easy direction:
hSep ≥ 2->4 norm
M = 𝔼i ai aiT ⊗ ai aiT
ρ=xx* ⊗ xx*
Harder direction:
2->4 norm ≥ hSep
Given an arbitrary M, can we make it look like 𝔼i aiai* ⊗ aiai* ?
Answer: yes, using techniques of [H, Montanaro; 1001.0017]
the dream
SSE
hardness
2->4
hSep
algorithms
…quasipolynomial (=exp(polylog(n)) upper and lower bounds for unique games
progress so far
SSE hardness??
1. Estimating hSep(M) ± 0.1 for n-dimensional M is at least
as hard as solving 3-SAT instance of length ≈log2(n).
[H.-Montanaro 1001.0017] [Aaronson-Beigi-Drucker-Fefferman-Shor 0804.0802]
2. The Exponential-Time Hypothesis (ETH) implies a lower
bound of Ω(nlog(n)) for hSep(M).
3. ∴ lower bound of Ω(nlog(n)) for estimating ||A||2->4 for
some family of projectors A.
4. These A might not be P≥λ for any graph G.
5. (Still, first proof of hardness for
constant-factor approximation of ||¢||24).
positive results about hierarchies:
1. use dual
Primal: max L[f(x)] over L such that
L is a linear map from deg ≤k polys to R
L[1] = 1
L[p(x) (∑i xi2 – 1)] = 0
L[p(x)2] ≥ 0
Dual: min λ such that
f(x) + p(x) (𝔼i xi2 – 1) + ∑i qi(x)2 = λ
for some polynomials p(x), {qi(x)} s.t. all degrees are ≤ k.
Interpretation: “Prove that f(x) is ≤ λ using only the
facts that 𝔼i xi2 – 1 = 0 and sum of square (SOS)
polynomials are ≥0. Use only terms of degree ≤k. ”
“Positivestellensatz” [Stengel ’74]
SoS proof example
z2≤z $ 0≤z≤1
Axiom: z2 ≤ z
Derive: z ≤ 1
1 – z = z – z2 + (1-z)2
≥ z – z2
(non-negativity of squares)
≥0
(axiom)
SoS proof of hypercontractivity
Hypercontractive inequality:
Let f:{0,1}nR be a polynomial of degree ≤d. Then
||f||4 ≤ 9d/4 ||f||2.
equivalently:
||Pd||2->4 ≤ 9d/4 where Pd projects onto deg ≤d polys.
Proof:
uses induction on n and Cauchy-Schwarz.
Only inequality is q(x)2 ≥ 0.
Implication: SDP returns answer ≤9d/4 on input Pd.
SoS proofs of UG soundness
[BBHKSZ ‘12]
Result: Degree-8 SoS relaxation refutes UG instances
based on long-code and short-code graphs
Proof: Rewrite previous soundness proofs as SoS proofs.
Ingredients:
1. Cauchy-Schwarz / Hölder
2. Hypercontractive inequality
3. Influence decoding
4. Independent rounding
5. Invariance principle
UG Integrality Gap:
Feasible SDP solution
SoS upper bound
Upper bound to actual solutions
actual solutions
positive results about
hierarchies: 2. use q. info
[Brandão-Christandl-Yard ’10] [Brandão-H. ‘12]
Idea:
Monogamy relations for entanglement imply performance
bounds on the SoS relaxation.
Proof sketch:
ρis k-extendable, lives on AB1 … Bk.
M can be implemented by measuring Bob, then Alice. (1-LOCC)
Let measurement outcomes be X,Y1,…,Yk.
Then
log(n) ≥ I(X:Y1…Yk) = I(X:Y1) + I(X:Y2|Y1) + … + I(X:Yk|Y1…Yk-1)
…algebra…
hSep(M) ≤ hk-ext(M) ≤ hSep(M) + c(log(n) / k)1/2
Alternate perspective
For i=1,…,k
• Measure Bi.
• If entropy of A doesn’t change, then A:Bi are ≈product.
• If entropy of A decreases, then condition on Bi.
the dream: quantum proofs for
classical algorithms
1. Information-theory proofs of de Finetti/monogamy,
e.g. [Brandão-Christandl-Yard, 1010.1750] [Brandão-H., 1210.6367]
hSep(M) ≤ hk-Ext(M) ≤ hSep(M) + (log(n) / k)1/2 ||M||
if M∈1-LOCC
2. M = ∑i aiai* aiai* is ∝ 1-LOCC.
3. Constant-factor approximation in time nO(log(n))?
4. Problem: ||M|| can be ≫ hSep(M). Need multiplicative
approximaton or we lose dim factors.
5. Still yields subexponential-time algorithm.
SDPs in quantum information
1. Goal: approximate Sep
Relaxation: k-extendable + PPT
2. Goal: λmin for Hamiltonian on n qudits
Relaxation: L : k-local observables  R
such that L[X†X] ≥ 0 for all k/2-local X.
3. Goal: entangled value of multiplayer games
Relaxation: L : products of ≤k operators  R
such that L[p†p] ≥ 0 ∀noncommutative poly p of degree ≤ k,
and operators on different parties commute.
Non-commutative positivstellensatz [Helton-McCullough ‘04]
relation between these? tools to analyze?
questions
We are developing some vocabulary for understanding
these hierarchies (SoS proofs, quantum entropy, etc.).
Are these the right terms?
Are they on the way to the right terms?
Unique games, small-set expansion, etc:
quasipolynomial hardness and/or algorithms
Relation of different SDPs for quantum states.
More tools to analyze #2 and #3.
Why is this an SDP?